Commentary on Proposition X.82

Back to Proposition X.82.

[Folio 0177 | p. 173]

Suppose, if possible, that, with the usual notation,

$$x - y = x' - y' ;$$

and let $x$ (say) be greater than $x'$.

In this case $(x^2 + y^2)$, $(x'^2 + y'^2)$ are both rational areas, and $2xy$, $2x'y'$ are both medial areas.

But, as before,

$$(x^2 + y^2) - (x'^2 + y'^2) = 2xy - 2x'y',$$

so that the difference between two medial areas is rational : which is impossible [x. 26].

Therefore etc.